{"id":"c241caee-b06f-4148-a3fc-5f23ecacc558","arxiv_id":"2505.17853","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Cyclic branched covers of complex hyperbolic manifolds have non-complex-hyperbolic Chern number ratios, proven exactly in dimension 2 and claimed with a gap in higher even dimensions.","lead":"This paper computes Chern number ratios of complex hyperbolic branched covers and proves they differ from complex hyperbolic manifolds in complex dimension 2, with a gap in the higher even dimensional argument. It gives a negative answer to a question of Deraux and Seshadri in dimension 2 and shows the author's earlier almost 1/4-pinched metric is not Kähler.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 (n=2) is sound, but Theorem 1.1 does not follow: the proof of (3.10) never controls the d-dependence of the cover (M',N'), so the 'finitely many d' conclusion is unsupported.","rationale":"The n=2 theorem survives scrutiny: equation (3.8) follows directly from the signature formula, \\chi(X)=d\\chi(M')-(d-1)\\chi(N'), and e((N')^\\perp)=\\chi(N')/2, and \\chi(N)\\neq 0 by Chern-Gauss-Bonnet, so c_1^2-3c_2\\neq 0 for every d\\geq 2. This is a genuine, parameter-free result and already gives a negative answer to the Deraux-Seshadri question in dimension 2 via [15]. The problem is Theorem 1.1. The proof reduces the hypothesis 'all Chern ratios equal to CP^n' to equation (3.10), but the reduction to a rational function of d with fixed coefficients is not justified. The cover (M',N') is chosen after d, and nothing prevents m, f_i(m), or c_r((N'_r)^\\perp) from depending on d; indeed the Stover-Toledo construction produces covers that do depend on d. Thus the assertion 'at most finitely many d' can fail, because a sum with d-dependent coefficients can vanish at every d. This is exactly the weakest assumption the reader identified, and it is load-bearing because Theorem 1.1 and the n>2 parts of Corollaries 1.5 and 1.6 rest on it. I also note a secondary issue in the proof of Corollary 1.5: closeness of Chern ratios controls the normalized difference (3.10)/\\chi(X), not the unnormalized expression, so the claim that the right side 'would approach 0' needs an additional scale-invariance argument. The appropriate verdict is unchanged: reject as written, while crediting the n=2 theorem and its corollaries as salvageable content.","tokens_in":9107,"tokens_out":19948,"duration_ms":216229,"concrete_test":"Fix n=4 and write the n=4 case of (3.10) as F_d = m_d(d-1)/5 \\chi(N) - (d^2-1)/(3d) A_1(d) + (d^2-1)(d^2-4)/(45d) A_2(d), where A_i(d)=c_i((N'_{i,d})^\\perp). For the Stover-Toledo covers used in Corollary 1.5, compute m_d, A_1(d), A_2(d) for d=2,3,5,7,11,... and check whether F_d vanishes for infinitely many d. Then test the 'arbitrary cover' quantifier by producing a second admissible cover with the same d and same degree m but different A_i(d), e.g. by pulling back through an additional etale cover of N'_d. If F_d=0 for infinitely many d, or if the coefficients can be varied at fixed d and m, the proof's claim that the coefficients are functions of m alone is refuted. If no such examples exist, the concern reduces to an omitted justification that may be repairable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 1.1, the cover (M',N') is chosen after d and is only required to have [N'] d-divisible. Hence m, f_i(m), and c_r((N'_r)^\\perp) are functions of d and of the choice of cover, not constants. The sentence after (3.10), 'since the Euler characteristic and all Chern numbers are independent of d', is therefore a quantifier error: a finite sum \\sum a_j(d)P_j(d) with d-dependent coefficients can vanish for infinitely many d even if no fixed-coefficient polynomial is identically zero. The proof gives no bound relating these coefficients to d or to m, so it does not rule out infinitely many d each with a cover for which the branched cover satisfies (3.10). Moreover, writing f_i(m) is unjustified: the auxiliary submanifolds N'_r are defined via transverse perturbations inside X, and their Chern classes are not shown to be determined by the degree m alone or to descend from a fixed submanifold of the base. Thus Theorem 1.1 is not established, and Corollaries 1.5 and 1.6 in dimensions n>2 inherit the gap. The n=2 calculation (3.8) is an independent, explicit, and correct argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Chern number ratios of cyclic branched covers of complex hyperbolic manifolds. For complex dimension n=2, it proves an explicit formula, c1^2(X)-3c2(X)=m(d-1)^2/(2d) χ(N)≠0, showing that the branched cover X is not complex hyperbolic. For arbitrary even n≥2, it claims (Theorem 1.1) that for all but finitely many d, every d-fold cyclic branched cover obtained from a finite cover (M',N') with [N'] d-divisible has at least one Chern number ratio different from the corresponding ratio for complex hyperbolic manifolds. The paper then derives two corollaries: a negative answer to a question by Deraux and Seshadri, and the non-Kählerity of the author's previously constructed almost 1/4-pinched metric in higher dimensions.","tokens_in":9191,"tokens_out":4024,"duration_ms":32991,"significance":"If Theorem 1.1 were established, the paper would resolve a natural question about pinching and Chern number rigidity and would strengthen the author's earlier construction. The n=2 result is a clean, explicit, and apparently correct calculation that already settles the question in real dimension four. The higher-dimensional statement, however, is not proven by the argument given: the proof of Theorem 1.1 contains a quantifier error concerning the dependence of the auxiliary finite covers on the branching degree d. Because this gap affects the main theorem and the higher-dimensional corollaries, the paper in its present form does not support its advertised even-dimensional claim.","major_comments":[{"comment":"The assertion that the left-hand side of (3.10) can vanish for at most finitely many d is unsupported. The cover (M',N') is chosen after d and is only required to make [N'] d-divisible; hence the degree m, the functions f_i(m), and the Chern classes c_r((N'_r)^\\perp) are all functions of d and of the choice of cover. A finite sum of the form Σ a_j(d) P_j(d) with d-dependent coefficients can vanish for infinitely many d even if no fixed-coefficient polynomial is identically zero. The sentence 'since the Euler characteristic and all Chern numbers are independent of d' is therefore a quantifier error: it conflates independence of d for a fixed cover with independence across a family of covers that is allowed to vary with d.","section":"§3.2, proof of Theorem 1.1, after Eq. (3.10)"},{"comment":"The rewriting of the Chern classes as f_i(m)c_i(N^\\perp_i) is not justified. The submanifolds N'_r are defined via transverse perturbations of Y inside the branched cover X, and their Chern classes are not shown to be determined by the degree m alone, nor are the N'_r shown to descend from fixed submanifolds of the base pair (M,N). Thus the expression in (3.10) is not a polynomial in d with constant coefficients, and the 'at most finitely many d' conclusion does not follow.","section":"§3.2, Eq. (3.10)"},{"comment":"The proof of Corollary 1.5 relies on the statement that the right-hand side of (3.10), with m_k substituted for m, approaches ±∞ as k→∞. However, the degree m_k and the Chern classes in (3.10) depend on k in an uncontrolled way, and the sign of the leading term is not established. For n>2 this does not prove that the expression stays away from zero as k→∞. Only in the n=2 case, where the explicit formula (3.8) gives a positive multiple of m_k(d-1)^2χ(N), is the divergence clear.","section":"§3.2, proof of Corollary 1.5"}],"minor_comments":[{"comment":"The remark states that equation (3.8) proves Corollaries 1.5 and 1.6 for n=2; this is correct, but the wording could be clarified to indicate that the higher-dimensional cases are not covered by the explicit computation.","section":"Remark 3.4"},{"comment":"The proof of Corollary 2.3 assumes that all Chern number ratios of M are equal to those of CP^n. The negation of Theorem 1.1 only requires the existence of at least one ratio that is equal, so the contradiction setup in Theorem 1.1 is stronger than needed; this is not an error but could be noted for clarity.","section":"§2, Corollary 2.3"},{"comment":"The phrase 'the ratio of Chern numbers' is sometimes used in the singular and sometimes in the plural; the paper would benefit from a consistent convention, e.g., 'some ratio' vs. 'all ratios'.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The n=2 theorem and its explicit formula (3.8) are correct and valuable, and the paper could be salvaged by refocusing on that result or by substantially strengthening the argument for higher n. However, the main theorem as stated (Theorem 1.1) and the higher-dimensional corollaries rest on an unproved quantifier step: the auxiliary covers and their Chern data are allowed to depend on d, so the 'finitely many d' conclusion does not follow. This is a load-bearing gap, not a local fix, and in my view the manuscript in its current form does not meet the standard for publication as a major theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Theorem 1.2 (n=2) is solid and worth knowing. Theorem 1.1 for n>2 is not established as written.\n\nThe n=2 part is a genuine contribution. The explicit formula c1^2(X) minus 3c2(X) equals m(d-1)^2/(2d) times chi(N) for d-fold cyclic branched covers of finite covers of (CH^2, CH^1) pairs is new, correctly derived, and has real consequences: it gives a negative answer to the Deraux-Seshadri question in dimension 2 and shows the author's almost 1/4-pinched metric is not Kähler there. The use of Hirzebruch's signature formula for branched covers, with Viro's correction, is apt. This part deserves publication.\n\nThe soft spot is exactly where the reader and stress-test put it: the proof of Theorem 1.1. After deriving equation (3.10), the author says it is clear that the equation holds for at most finitely many d. That is not clear, and it is probably false as a logical step. The cover (M',N') is chosen after d, so the degree m, the submanifolds N'_r, and the Chern classes c_r((N'_r)^perp) all depend on d. A finite sum of d-dependent coefficients times d-dependent terms can vanish for infinitely many d without any fixed polynomial being identically zero. The proof gives no bound relating these quantities to d. Additionally, writing f_i(m) is unjustified: the N'_r are constructed via transverse perturbations inside X, and their Chern classes are not shown to be determined by the base degree m alone or to descend from fixed submanifolds of the base. So the 'for all but finitely many d' conclusion for arbitrary covers is unsupported. Corollaries 1.5 and 1.6 for n>2 inherit this gap.\n\nThe paper's own remarks concede the difficulty: Remark 1.4 attributes the finiteness condition to the difficulty of exact signature computations, and Remark 3.2 says exact values for the relevant Chern classes seem difficult to calculate. That is not a minor technical caveat; it is the load-bearing step of Theorem 1.1.\n\nFor whom is this paper? Anyone working on complex hyperbolic branched covers, Chern number rigidity, or the Deraux-Seshadri question should know the n=2 result. The n>2 theorem is a plausible conjecture, but the proof needs real work. A serious referee could help the author either prove the needed nonvanishing with d-dependence controlled or restrict the statement to settings where the cover is fixed.\n\nRecommendation: send to peer review. The n=2 theorem is worth publishing even if Theorem 1.1 is not yet proven; the referee should push for a corrected or weakened general statement.","headline":"The n=2 theorem is a clean, correct result that answers Deraux-Seshadri in dimension 2; the even-dimensional theorem as written has a quantifier gap around d-dependent covers.","tokens_in":801,"tokens_out":1162,"would_cite":true,"duration_ms":27188,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55R25","57R20","53C20","53C24"],"pacs":[],"model":"deepseek-v4-flash","headline":"Branched covers of complex hyperbolic manifolds have non-hyperbolic Chern ratios.","keywords":["complex hyperbolic manifolds","Chern numbers","Chern number ratios","cyclic branched covers","signature of branched covers","proportionality theorem","almost 1/4-pinched metrics","Kähler manifolds"],"falsifier":"Compute the signatures $\\Sigma(Y_{2r})$ or the Chern classes $c_r((N'_r)^\\perp)$ for a concrete family of cyclic branched covers in even complex dimension $n\\ge 4$, and check whether the expression in equation (3.10) vanishes for infinitely many $d$; a single family where it vanishes identically would refute the 'all but finitely many $d$' conclusion of Theorem 1.1.","tokens_in":8684,"feed_emoji":"📐","tokens_out":12513,"duration_ms":83999,"temperature":0.7,"pith_summary":"This paper proves that the $d$-fold cyclic branched covers of complex hyperbolic manifolds introduced through divisibility of the branch locus have Chern-number ratios that are not all equal to the complex hyperbolic ratios. In every even complex dimension $n$, for all but finitely many branching degrees $d$, an arbitrary finite cover of a pair modeled on $(\\mathbb{CH}^n,\\mathbb{CH}^{n-1})$ produces a branched cover $X$ with some Chern-number ratio different from the corresponding ratio of a complex hyperbolic manifold. In complex dimension $2$ the statement is unconditional: for every $d\\ge 2$, the identity $c_1^2(X)-3c_2(X)=\\frac{m(d-1)^2}{2d}\\chi(N)\\neq 0$ holds, where $m$ is the degree of the finite cover and $\\chi(N)$ is the Euler characteristic of the branch hypersurface. Because closed complex hyperbolic surfaces satisfy $c_1^2=3c_2$, this separates the branched covers from complex hyperbolic manifolds by an explicit nonzero amount. The result answers a motivating question about almost $1/4$-pinched K\\\"ahler metrics in the negative and implies that an earlier almost $1/4$-pinched metric on these manifolds is not K\\\"ahler.","feed_headline":"Branched covers dodge the complex hyperbolic Chern ratio","feed_subtitle":"In dimension 2 the gap is a nonzero multiple of the branch locus Euler characteristic, growing with the cover.","key_machinery":"The mechanism is the interaction of three classical formulas: the proportionality theorem for Chern numbers, which makes every Chern number of a closed complex hyperbolic manifold a fixed multiple of the corresponding Chern number of complex projective space, so all ratios are equal to the projective ratios; the signature theorem, which expresses the signature as a polynomial in Pontrjagin and Chern numbers; and the signature formula for cyclic branched covers, which writes the signature of $X$ in terms of the signature of the base and the signatures of transverse self-intersections of the branch locus through the rational function $\\mathrm{sign}(t)=\\frac{(1+t)^d+(1-t)^d}{(1+t)^d-(1-t)^d}\\,t$. In dimension $2$ the latter collapses to $\\Sigma(X)=d\\Sigma(M')-\\frac{d^2-1}{6d}\\chi(N')$, and together with $\\chi(X)=d\\chi(M')-(d-1)\\chi(N')$ and $\\Sigma(M')=\\chi(M')/3$ this yields the exact identity $c_1^2(X)-3c_2(X)=\\frac{m(d-1)^2}{2d}\\chi(N)$ in the paper.","core_discovery":"The paper's central claim is that branched covers built this way escape the rigidity of Chern-number ratios for complex hyperbolic manifolds. For even $n$, the claim is that for all but finitely many branching degrees $d$, no matter which finite cover $(M',N')$ is chosen with $[N']$ $d$-divisible, the $d$-fold cyclic branched cover $X$ has at least one Chern-number ratio different from the corresponding ratio of a complex hyperbolic manifold. The $n=2$ case is stronger: for every $d\\ge 2$, $c_1^2(X)-3c_2(X)=\\frac{m(d-1)^2}{2d}\\chi(N)\\neq 0$, so the only ratio $c_1^2/c_2$ is not $3$. The nonzero discrepancy is a consequence of the signature of the branched cover and grows with the degree of the preliminary cover.","pith_inferences":["An extension the paper leaves implicit: the exact $n=2$ formula gives a quantitative gap, so any K\\\"ahler metric on $X$ with Chern ratio within $\\epsilon$ of $3$ must come from a cover with bounded degree or a branch locus with small Euler characteristic.","A testable extension: in higher even dimensions the same finite signature expansion should yield an explicit polynomial in $d$ and $m$ once the Chern classes $c_r((N'_r)^\\perp)$ are computed; the author notes that such values would likely show the expression is never zero for all $d$.","The method is tied to even dimensions because self-intersection signatures vanish in odd complex dimensions, so an odd-dimensional analogue would need a different invariant or a computation of intersection signatures, as a three-dimensional calculation cited in the paper suggests."],"forward_implications":["For $n=2$, every such branched cover has $c_1^2/c_2\\neq 3$, and the gap $c_1^2-3c_2$ grows with the degree of the preliminary cover.","The motivating question about almost $1/4$-pinched K\\\"ahler metrics is answered negatively: compact K\\\"ahler manifolds exist with Chern-number ratios bounded away from the complex hyperbolic ratios while admitting metrics arbitrarily close to $1/4$-pinched.","The almost $1/4$-pinched Riemannian metric constructed earlier on these branched covers cannot be K\\\"ahler.","In even dimensions, taking larger finite covers does not make the branched covers resemble complex hyperbolic manifolds in their Chern-number ratios; for large $d$ the discrepancy in $n=2$ actually grows."],"supporting_citations":[{"why":"Supplies the proportionality theorem: all Chern numbers of a closed complex hyperbolic manifold are fixed multiples of the corresponding Chern numbers of complex projective space, so all Chern-number ratios match the projective ratios.","marker":"[8]"},{"why":"Supplies the signature formula for cyclic branched covers, expressing the signature of the cover through the given rational function and signatures of self-intersections of the branch locus.","marker":"[9]"},{"why":"Supplies the signature theorem used to express the signature as a linear combination of Pontrjagin and Chern numbers, yielding Corollary 2.3.","marker":"[11]"},{"why":"Supplies the characteristic-class background used in the calculations: Chern-to-Pontrjagin formulas, $p_1=c_1^2-2c_2$, and the signature formula $\\Sigma=p_1/3$ in dimension 4.","marker":"[14]"},{"why":"Corrects the general signature formula in [9] and gives the version used to derive equation (3.2).","marker":"[20]"},{"why":"Gives the identity $e((N')^\\perp)=\\chi(N')/2$ for a totally geodesic complex hypersurface, which turns the dimension-2 signature formula into equation (3.5).","marker":"[5]"},{"why":"Provides the intersection-theory fact that identifies the signature of the self-intersection $Y_{2r}$ with the top Chern number of the normal bundle of $Y_r$, used in the proof of Theorem 1.1.","marker":"[4]"},{"why":"Formulates the motivating question about almost $1/4$-pinched K\\\"ahler metrics and Chern-number ratios that the paper answers in the negative.","marker":"[3]"},{"why":"Constructs the almost $1/4$-pinched metrics on these branched covers whose non-K\\\"ahler nature is established in the corollaries.","marker":"[15]"}],"fun_headline_variants":["Even-dim branched covers break complex hyperbolic Chern ratios","Chern ratios of branched covers differ in even dimensions","Branched covers: Chern gap in even dims, explicit in 2D","Non-hyperbolic Chern ratios from branched covers","Branched covers shift Chern numbers away from hyperbolic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The general even-dimensional theorem depends on the unproved assertion that the rational function of $d$ in equation (3.10) is not identically zero, despite the cover, the degree $m$, and the Chern classes of the submanifolds all being allowed to depend on $d$.","fun_headline_variants_meta":{"raw":{"variants":["Even-dim branched covers break complex hyperbolic Chern ratios","Chern ratios of branched covers differ in even dimensions","Branched covers: Chern gap in even dims, explicit in 2D","Non-hyperbolic Chern ratios from branched covers","Branched covers shift Chern numbers away from hyperbolic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001101,"raw_usage":{"total_tokens":4516,"prompt_tokens":791,"completion_tokens":3725,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":407,"completion_tokens_details":{"reasoning_tokens":3644}},"tokens_in":407,"tokens_out":3725,"duration_ms":23023,"temperature":1.0,"reasoning_tokens":3644,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:41:06.665613+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the signatures $\\Sigma(Y_{2r})$ or the Chern classes $c_r((N'_r)^\\perp)$ for a concrete family of cyclic branched covers in even complex dimension $n\\ge 4$, and check whether the expression in equation (3.10) vanishes for infinitely many $d$; a single family where it vanishes identically would refute the 'all but finitely many $d$' conclusion of Theorem 1.1.","supporting_citations":[{"cited_title":"Automorphe Formen und der Satz vo n Riemann-Roch","cited_arxiv_id":null,"evidence_quote":"Supplies the proportionality theorem: all Chern numbers of a closed complex hyperbolic manifold are fixed multiples of the corresponding Chern numbers of complex projective space, so all Chern-number ratios match the projective ratios."},{"cited_title":"The signature of ramified coverin gs","cited_arxiv_id":null,"evidence_quote":"Supplies the signature formula for cyclic branched covers, expressing the signature of the cover through the given rational function and signatures of self-intersections of the branch locus."},{"cited_title":"Topological methods in algebraic geometry","cited_arxiv_id":null,"evidence_quote":"Supplies the signature theorem used to express the signature as a linear combination of Pontrjagin and Chern numbers, yielding Corollary 2.3."},{"cited_title":"Milnor and James D","cited_arxiv_id":null,"evidence_quote":"Supplies the characteristic-class background used in the calculations: Chern-to-Pontrjagin formulas, $p_1=c_1^2-2c_2$, and the signature formula $\\Sigma=p_1/3$ in dimension 4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Corrects the general signature formula in [9] and gives the version used to derive equation (3.2)."},{"cited_title":"Goldman, Michael Kapovich, and Bernhard Leeb","cited_arxiv_id":null,"evidence_quote":"Gives the identity $e((N')^\\perp)=\\chi(N')/2$ for a totally geodesic complex hypersurface, which turns the dimension-2 signature formula into equation (3.5)."},{"cited_title":"Intersection theory, volume 2 of Ergeb","cited_arxiv_id":null,"evidence_quote":"Provides the intersection-theory fact that identifies the signature of the self-intersection $Y_{2r}$ with the top Chern number of the normal bundle of $Y_r$, used in the proof of Theorem 1.1."},{"cited_title":"Almost quarter-pinc hed K¨ ahler metrics and Chern num- bers","cited_arxiv_id":null,"evidence_quote":"Formulates the motivating question about almost $1/4$-pinched K\\\"ahler metrics and Chern-number ratios that the paper answers in the negative."},{"cited_title":"K\\\"{a}hler manifolds with an almost $1/4$-pinched metric","cited_arxiv_id":"2307.15550","evidence_quote":"Constructs the almost $1/4$-pinched metrics on these branched covers whose non-K\\\"ahler nature is established in the corollaries."}],"review_version":1}